A geometric study of Wasserstein spaces: ultrametrics

Benoît Kloeckner (1)

1 IF - Institut Fourier

We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L^p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of l^1. As a consequence, it is connected by 1/p-Hölder arcs, but any a-Hölder arc with a>1/p must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when X is ultrametric; howeverthanks to the Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate that needed a strong rectifiability assumption. The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of \emph{regular} ultrametric spaces, i.e. spaces of the form {1,...,k}^N with a natural ultrametric. We are also lead to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.

Mots clés: Wasserstein spaces; ultrametric spaces; Hausdorff dimension; Minkowski dimension; Box dimension; Optimal transportation; bi-Lipschitz geometry.; ultrametric skeleton; critical parameter; bi-Lipschitz geometry

Domaines: Analyse fonctionnelle [math.FA]; Géométrie métrique [math.MG]; Géométrie différentielle [math.DG]

Benoît Kloeckner. A geometric study of Wasserstein spaces: ultrametrics. Mathematika, 2015, 61 (01), pp.162-178. ⟨10.1112/S0025579314000059⟩. ⟨hal-00815396v2⟩